# Full text of "Treatise On Analysis Vol-Ii"

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```18   CANONICAL DECOMPOSITIONS OF A MEASURE       205

mapping U is said to be of type (p, q) if U(E) c LJ(Y, v) and if U is continuous in the
topology induced on E by that of L£(X, /x), U being considered as a mapping of E
into L£(Y, v). Let \\U\\ptq denote the norm of this linear mapping. If p =£ +00, then
U may be extended by continuity to a linear mapping, with norm \\U\\pt q , of Lg(X, JJL)
into LJ(Y, v). The same is true if p = 4- oo, provided that p, is bounded.

Show that if U is of type (p0 , q0) and also of type Oi, #0, then £/ is also of type
(p, q) where

1 __ 1 -t      t          1 _ 1 -t      t

P      Po       Pi'        q       Qo       <?i

for all t e [0, 1] (with the convention that l/oo =0, 1/0= +00); moreover, we have

\\u\\,. 9z ii oiij-v iiWpi.«i

(M. Riesz-Thorin interpolation theorem). (Use Problem 1, by majorizing the integral
\ \(U '/)#! dv for /e E and g e L£(Y", v), where - + — = 1 ; for this purpose, put

J                                                                                                                                                ~l         T

/= I/IK, 0= 101" with H = \v\ = 1, and /c= |/|;w, ^c= \g\^v for all complex
numbers £. Then choose appropriately two affine-linear functions

and apply the three-line theorem (Problem 6) to the holomorphic function

18. CANONICAL DECOMPOSITIONS OF A MEASURE

Two (complex) measures ju, v on X are said to be disjoint if inf(|^|, |v|) = 0.
A. measure JJL is said to be concentrated on a set M, or carried by M, if X — M
is \^-negligible, or equivalent^ if |ju| = <pM • |ju| (13.15.3); every measure
with base |ju| is then also concentrated on M (13.15.5).

(13.18.1) For two measures ju, v to be disjoint it is necessary and sufficient that
there should exist two disjoint subsets M, N ofX such that ju is concentrated on
M and v is concentrated on N; and the sets M, N can be chosen to be universally
measurable.

We may restrict ourselves to the case where ^ and v are positive, and then
we can write JJL — g * p and v = h • p, where p is a positve measure and g, h are
locally p-integrable functions ;>0 ((13.15.2) and (13.15.3)). We have then
inf(/i, v) = inf(#, h) - p, and hence inf^u, v) = 0 if and only if inf(#, h) is
p-negligible (13.15.3). If M0 and N0 are the sets of points xeX such that
g(x) ^ 0 and h(x) ^ 0, respectively, then inf(#, h) is p-negligible if and onlygmen-Lindelof principle as in Section 9.5, Problem 17, by taking g(s) = e*2.)
```